Subgroups — C₁₀₉×#11232 #42129226

Order 27904.

870 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#107×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#340×C₁₀₉	C₂	1	2	yes	maximal
4	C₂×#340×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₁₀₉×C₄⋊#95	C₂	1	2	yes	maximal
6	C₂³×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
9	C₂²×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
10	C₂²×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×#107×C₁₀₉	—	2	2	no	—
12	C₂×#107×C₁₀₉	—	2	4	no	—
13	C₂×C₄⋊C₈×C₁₀₉	—	2	2	no	—
14	C₂×C₄⋊C₈×C₁₀₉	—	2	2	no	—
15	C₂×C₄⋊C₈×C₁₀₉	C₄	1	4	yes	complemented
16	C₂×#95×C₁₀₉	—	2	2	no	—
17	C₂×#95×C₁₀₉	—	2	2	no	—
18	#340×C₁₀₉	—	2	2	no	—
19	#340×C₁₀₉	—	2	4	no	—
20	C₂³×C₄×C₁₀₉	—	2	2	no	—
21	C₂³×C₄×C₁₀₉	—	2	2	no	—
22	C₂³×C₄×C₁₀₉	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₀₉	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
25	C₂²×C₈×C₁₀₉	—	2	2	no	—
26	C₂²×C₈×C₁₀₉	—	2	2	no	—
27	C₂²×C₈×C₁₀₉	—	2	2	no	—
28	C₂²×C₈×C₁₀₉	—	2	1	no	—
29	C₂²×C₈×C₁₀₉	—	2	1	no	—
30	C₂²×C₈×C₁₀₉	—	2	2	no	—
31	C₂²×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₁₀₉	D₄	1	2	yes	—
34	C₂²×C₈×C₁₀₉	D₄	1	2	yes	—
35	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
36	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
37	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
38	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
39	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
40	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
41	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
42	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
43	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
44	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
45	C₂×C₄⋊C₄×C₁₀₉	D₄	1	4	yes	—
46	C₂×C₄⋊C₄×C₁₀₉	Q₈	1	4	yes	—
47	C₂×C₄²×C₁₀₉	—	2	1	no	—
48	C₂×C₄²×C₁₀₉	—	2	1	no	—
49	C₂×C₄²×C₁₀₉	—	2	1	no	—
50	C₂×C₄²×C₁₀₉	—	2	1	no	—
51	C₂×C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₁₀₉	Q₈	1	2	yes	—
53	#107×C₁₀₉	—	2	8	no	—
54	C₄⋊C₈×C₁₀₉	—	2	8	no	—
55	#95×C₁₀₉	—	4	4	no	—
56	#95×C₁₀₉	—	4	4	no	—
57	C₂⁴×C₁₀₉	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₀₉	—	2	1	no	—
59	C₂²×C₄×C₁₀₉	—	2	1	no	—
60	C₂²×C₄×C₁₀₉	—	2	1	no	—
61	C₂²×C₄×C₁₀₉	—	2	1	no	—
62	C₂²×C₄×C₁₀₉	—	2	1	no	—
63	C₂²×C₄×C₁₀₉	—	2	1	no	—
64	C₂²×C₄×C₁₀₉	—	2	2	no	—
65	C₂²×C₄×C₁₀₉	—	2	2	no	—
66	C₂²×C₄×C₁₀₉	—	2	4	no	—
67	C₂²×C₄×C₁₀₉	—	2	4	no	—
68	C₂²×C₄×C₁₀₉	—	4	2	no	—
69	C₂²×C₄×C₁₀₉	—	4	2	no	—
70	C₂²×C₄×C₁₀₉	—	4	2	no	—
71	C₂²×C₄×C₁₀₉	—	4	2	no	—
72	C₂²×C₄×C₁₀₉	—	4	2	no	—
73	C₂²×C₄×C₁₀₉	—	4	2	no	—
74	C₂²×C₄×C₁₀₉	—	4	2	no	—
75	C₂²×C₄×C₁₀₉	—	4	2	no	—
76	C₂²×C₄×C₁₀₉	—	4	2	no	—
77	C₂²×C₄×C₁₀₉	—	4	2	no	—
78	C₂²×C₄×C₁₀₉	—	4	2	no	—
79	C₂²×C₄×C₁₀₉	—	4	2	no	—
80	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
81	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
82	C₂²×C₄×C₁₀₉	D₈	1	2	yes	—
83	C₂²×C₄×C₁₀₉	Q₁₆	1	2	yes	—
84	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
89	C₂²×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
90	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
91	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
92	C₂²×C₄×C₁₀₉	#41	1	2	yes	—
93	C₂²×C₄×C₁₀₉	#41	1	2	yes	—
94	C₂×C₈×C₁₀₉	—	2	4	no	—
95	C₂×C₈×C₁₀₉	—	2	2	no	—
96	C₂×C₈×C₁₀₉	—	2	4	no	—
97	C₂×C₈×C₁₀₉	—	2	2	no	—
98	C₂×C₈×C₁₀₉	—	2	2	no	—
99	C₂×C₈×C₁₀₉	—	2	4	no	—
100	C₂×C₈×C₁₀₉	—	4	1	no	—
101	C₂×C₈×C₁₀₉	—	4	2	no	—
102	C₂×C₈×C₁₀₉	—	4	1	no	—
103	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	4	yes	—
104	C₄⋊C₄×C₁₀₉	—	2	4	no	—
105	C₄⋊C₄×C₁₀₉	—	2	4	no	—
106	C₄⋊C₄×C₁₀₉	—	2	4	no	—
107	C₄⋊C₄×C₁₀₉	—	2	4	no	—
108	C₄⋊C₄×C₁₀₉	—	4	2	no	—
109	C₄⋊C₄×C₁₀₉	—	4	2	no	—
110	C₄⋊C₄×C₁₀₉	—	4	2	no	—
111	C₄⋊C₄×C₁₀₉	—	4	2	no	—
112	C₄²×C₁₀₉	—	2	2	no	—
113	C₄²×C₁₀₉	—	2	2	no	—
114	C₄²×C₁₀₉	—	2	2	no	—
115	C₄²×C₁₀₉	—	2	2	no	—
116	C₂³×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂³×C₁₀₉	#121	1	1	yes	characteristic
118	C₂³×C₁₀₉	#121	1	1	yes	characteristic
119	C₂³×C₁₀₉	#121	1	2	yes	—
120	C₂³×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
121	C₂³×C₁₀₉	C₂×D₈	1	1	yes	characteristic
122	C₂³×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
123	C₂³×C₁₀₉	#135	1	1	yes	characteristic
124	C₂³×C₁₀₉	#135	1	1	yes	characteristic
125	C₂³×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
126	C₂³×C₁₀₉	#137	1	2	yes	—
127	C₂×C₄×C₁₀₉	—	2	2	no	—
128	C₂×C₄×C₁₀₉	—	2	2	no	—
129	C₂×C₄×C₁₀₉	—	2	2	no	—
130	C₂×C₄×C₁₀₉	—	2	2	no	—
131	C₂×C₄×C₁₀₉	—	2	2	no	—
132	C₂×C₄×C₁₀₉	—	2	2	no	—
133	C₂×C₄×C₁₀₉	—	2	2	no	—
134	C₂×C₄×C₁₀₉	—	2	4	no	—
135	C₂×C₄×C₁₀₉	—	4	2	no	—
136	C₂×C₄×C₁₀₉	—	4	1	no	—
137	C₂×C₄×C₁₀₉	—	4	1	no	—
138	C₂×C₄×C₁₀₉	—	4	4	no	—
139	C₂×C₄×C₁₀₉	—	4	4	no	—
140	C₂×C₄×C₁₀₉	—	4	4	no	—
141	C₂×C₄×C₁₀₉	—	4	4	no	—
142	C₂×C₄×C₁₀₉	—	4	4	no	—
143	C₂×C₄×C₁₀₉	—	4	4	no	—
144	C₂×C₄×C₁₀₉	—	8	2	no	—
145	C₂×C₄×C₁₀₉	—	8	2	no	—
146	C₂×C₄×C₁₀₉	—	8	2	no	—
147	C₂×C₄×C₁₀₉	—	8	2	no	—
148	C₂×C₄×C₁₀₉	—	8	2	no	—
149	C₂×C₄×C₁₀₉	—	8	2	no	—
150	C₂×C₄×C₁₀₉	—	8	2	no	—
151	C₂×C₄×C₁₀₉	—	8	2	no	—
152	C₂×C₄×C₁₀₉	#107	1	4	yes	complemented
153	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
154	C₂×C₄×C₁₀₉	#117	1	1	yes	characteristic
155	C₂×C₄×C₁₀₉	C₄×D₄	1	2	yes	—
156	C₂×C₄×C₁₀₉	C₄×Q₈	1	2	yes	—
157	C₂×C₄×C₁₀₉	#120	1	1	yes	characteristic
158	C₂×C₄×C₁₀₉	#121	1	2	yes	—
159	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
160	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
161	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
162	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
163	C₂×C₄×C₁₀₉	#123	1	1	yes	characteristic
164	C₂×C₄×C₁₀₉	#123	1	2	yes	—
165	C₂×C₄×C₁₀₉	#124	1	1	yes	characteristic
166	C₂×C₄×C₁₀₉	#125	1	2	yes	—
167	C₂×C₄×C₁₀₉	#125	1	2	yes	—
168	C₂×C₄×C₁₀₉	#126	1	2	yes	—
169	C₂×C₄×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
170	C₂×C₄×C₁₀₉	C₄⋊Q₈	1	2	yes	—
171	C₈×C₁₀₉	—	2	4	no	—
172	C₈×C₁₀₉	—	4	2	no	—
173	C₂²×C₁₀₉	#380	1	1	yes	characteristic
174	C₂²×C₁₀₉	#389	1	2	yes	—
175	C₂²×C₁₀₉	#394	1	1	yes	characteristic
176	C₂²×C₁₀₉	#396	1	2	yes	—
177	C₂²×C₁₀₉	#398	1	1	yes	characteristic
178	C₂²×C₁₀₉	C₂×#107	1	1	yes	characteristic
179	C₂²×C₁₀₉	#427	1	1	yes	characteristic
180	C₂²×C₁₀₉	#428	1	2	yes	—
181	C₂²×C₁₀₉	#459	1	2	yes	—
182	C₂²×C₁₀₉	#462	1	2	yes	—
183	C₂²×C₁₀₉	#463	1	2	yes	—
184	C₂²×C₁₀₉	#464	1	2	yes	—
185	C₂²×C₁₀₉	#466	1	1	yes	characteristic
186	C₂²×C₁₀₉	#466	1	1	yes	characteristic
187	C₂²×C₁₀₉	#467	1	1	yes	characteristic
188	C₂²×C₁₀₉	#467	1	1	yes	characteristic
189	C₂²×C₁₀₉	#468	1	2	yes	—
190	C₂²×C₁₀₉	#469	1	1	yes	characteristic
191	C₂²×C₁₀₉	#470	1	1	yes	characteristic
192	C₂²×C₁₀₉	#474	1	2	yes	—
193	C₂²×C₁₀₉	#477	1	2	yes	—
194	C₂²×C₁₀₉	#478	1	2	yes	—
195	C₂²×C₁₀₉	#479	1	2	yes	—
196	C₄×C₁₀₉	—	2	4	no	—
197	C₄×C₁₀₉	—	4	2	no	—
198	C₄×C₁₀₉	—	8	4	no	—
199	C₄×C₁₀₉	—	8	4	no	—
200	C₄×C₁₀₉	#386	1	1	yes	characteristic
201	C₄×C₁₀₉	#387	1	2	yes	—
202	C₄×C₁₀₉	#391	1	1	yes	characteristic
203	C₄×C₁₀₉	#395	1	1	yes	characteristic
204	C₄×C₁₀₉	#398	1	2	yes	—
205	C₄×C₁₀₉	#400	1	1	yes	characteristic
206	#11232	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
207	C₂×C₁₀₉	#1441	1	1	yes	characteristic
208	C₂×C₁₀₉	#1808	1	1	yes	characteristic
209	C₂×C₁₀₉	#1809	1	1	yes	characteristic
210	C₂×C₁₀₉	#1947	1	2	yes	—
211	C₂×C₁₀₉	#1949	1	2	yes	—
212	C₂×C₁₀₉	#2054	1	1	yes	characteristic
213	C₂×C₁₀₉	#2056	1	1	yes	characteristic
214	C₂×C₁₀₉	#2067	1	2	yes	—
215	C₂×C₁₀₉	#2068	1	2	yes	—
216	C₂×C₁₀₉	#2086	1	1	yes	characteristic
217	C₂×C₁₀₉	#2087	1	1	yes	characteristic
218	C₂²×#107	C₂×C₁₀₉	1	1	yes	characteristic
219	C₂²×C₄⋊C₈	C₂×C₁₀₉	1	1	yes	characteristic
220	C₂×#340	C₂×C₁₀₉	1	2	yes	—
221	C₂×#340	C₂×C₁₀₉	1	1	yes	characteristic
222	C₄⋊#95	C₂×C₁₀₉	1	2	yes	—
223	C₁₀₉	#11232	1	1	yes	characteristic, direct product factor, Sylow
224	C₂³×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
225	C₂³×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
226	C₂²×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
227	C₂²×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
228	C₂²×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
229	C₂×#107	—	2	2	no	—
230	C₂×#107	—	2	4	no	—
231	C₂×C₄⋊C₈	—	2	2	no	—
232	C₂×C₄⋊C₈	—	2	2	no	—
233	C₂×C₄⋊C₈	C₄×C₁₀₉	1	4	yes	complemented
234	C₂×#95	—	2	2	no	—
235	C₂×#95	—	2	2	no	—
236	#340	—	2	2	no	—
237	#340	—	2	4	no	—
238	C₂³×C₄	—	2	2	no	—
239	C₂³×C₄	—	2	2	no	—
240	C₂³×C₄	D₄×C₁₀₉	1	1	yes	characteristic
241	C₂³×C₄	D₄×C₁₀₉	1	1	yes	characteristic
242	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
243	C₂²×C₈	—	2	2	no	—
244	C₂²×C₈	—	2	2	no	—
245	C₂²×C₈	—	2	2	no	—
246	C₂²×C₈	—	2	1	no	—
247	C₂²×C₈	—	2	1	no	—
248	C₂²×C₈	—	2	2	no	—
249	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
250	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
251	C₂²×C₈	D₄×C₁₀₉	1	2	yes	—
252	C₂²×C₈	D₄×C₁₀₉	1	2	yes	—
253	C₂×C₄⋊C₄	—	2	2	no	—
254	C₂×C₄⋊C₄	—	2	2	no	—
255	C₂×C₄⋊C₄	—	2	2	no	—
256	C₂×C₄⋊C₄	—	2	2	no	—
257	C₂×C₄⋊C₄	—	2	2	no	—
258	C₂×C₄⋊C₄	—	2	2	no	—
259	C₂×C₄⋊C₄	—	2	2	no	—
260	C₂×C₄⋊C₄	—	2	2	no	—
261	C₂×C₄⋊C₄	—	2	2	no	—
262	C₂×C₄⋊C₄	—	2	2	no	—
263	C₂×C₄⋊C₄	D₄×C₁₀₉	1	4	yes	—
264	C₂×C₄⋊C₄	Q₈×C₁₀₉	1	4	yes	—
265	C₂×C₄²	—	2	1	no	—
266	C₂×C₄²	—	2	1	no	—
267	C₂×C₄²	—	2	1	no	—
268	C₂×C₄²	—	2	1	no	—
269	C₂×C₄²	C₂×C₄×C₁₀₉	1	2	yes	—
270	C₂×C₄²	Q₈×C₁₀₉	1	2	yes	—
271	#107	—	2	8	no	—
272	C₄⋊C₈	—	2	8	no	—
273	#95	—	4	4	no	—
274	#95	—	4	4	no	—
275	C₂⁴	C₂×D₄×C₁₀₉	1	1	yes	characteristic
276	C₂²×C₄	—	2	1	no	—
277	C₂²×C₄	—	2	1	no	—
278	C₂²×C₄	—	2	1	no	—
279	C₂²×C₄	—	2	1	no	—
280	C₂²×C₄	—	2	1	no	—
281	C₂²×C₄	—	2	1	no	—
282	C₂²×C₄	—	2	2	no	—
283	C₂²×C₄	—	2	2	no	—
284	C₂²×C₄	—	2	4	no	—
285	C₂²×C₄	—	2	4	no	—
286	C₂²×C₄	—	4	2	no	—
287	C₂²×C₄	—	4	2	no	—
288	C₂²×C₄	—	4	2	no	—
289	C₂²×C₄	—	4	2	no	—
290	C₂²×C₄	—	4	2	no	—
291	C₂²×C₄	—	4	2	no	—
292	C₂²×C₄	—	4	2	no	—
293	C₂²×C₄	—	4	2	no	—
294	C₂²×C₄	—	4	2	no	—
295	C₂²×C₄	—	4	2	no	—
296	C₂²×C₄	—	4	2	no	—
297	C₂²×C₄	—	4	2	no	—
298	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
299	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
300	C₂²×C₄	D₈×C₁₀₉	1	2	yes	—
301	C₂²×C₄	Q₁₆×C₁₀₉	1	2	yes	—
302	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
303	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
304	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
305	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
306	C₂²×C₄	C₂×Q₈×C₁₀₉	1	2	yes	—
307	C₂²×C₄	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
308	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
309	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
310	C₂²×C₄	#41×C₁₀₉	1	2	yes	—
311	C₂²×C₄	#41×C₁₀₉	1	2	yes	—
312	C₂×C₈	—	2	4	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	2	4	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	—	2	4	no	—
318	C₂×C₈	—	4	1	no	—
319	C₂×C₈	—	4	2	no	—
320	C₂×C₈	—	4	1	no	—
321	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	4	yes	—
322	C₄⋊C₄	—	2	4	no	—
323	C₄⋊C₄	—	2	4	no	—
324	C₄⋊C₄	—	2	4	no	—
325	C₄⋊C₄	—	2	4	no	—
326	C₄⋊C₄	—	4	2	no	—
327	C₄⋊C₄	—	4	2	no	—
328	C₄⋊C₄	—	4	2	no	—
329	C₄⋊C₄	—	4	2	no	—
330	C₄²	—	2	2	no	—
331	C₄²	—	2	2	no	—
332	C₄²	—	2	2	no	—
333	C₄²	—	2	2	no	—
334	C₂³	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
335	C₂³	#121×C₁₀₉	1	1	yes	characteristic
336	C₂³	#121×C₁₀₉	1	1	yes	characteristic
337	C₂³	#121×C₁₀₉	1	2	yes	—
338	C₂³	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
339	C₂³	C₂×D₈×C₁₀₉	1	1	yes	characteristic
340	C₂³	C₂×Q₁₆×C₁₀₉	1	1	yes	characteristic
341	C₂³	#135×C₁₀₉	1	1	yes	characteristic
342	C₂³	#135×C₁₀₉	1	1	yes	characteristic
343	C₂³	AGL(1,ℤ/8)×C₁₀₉	1	2	yes	—
344	C₂³	#137×C₁₀₉	1	2	yes	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	4	no	—
353	C₂×C₄	—	4	2	no	—
354	C₂×C₄	—	4	1	no	—
355	C₂×C₄	—	4	1	no	—
356	C₂×C₄	—	4	4	no	—
357	C₂×C₄	—	4	4	no	—
358	C₂×C₄	—	4	4	no	—
359	C₂×C₄	—	4	4	no	—
360	C₂×C₄	—	4	4	no	—
361	C₂×C₄	—	4	4	no	—
362	C₂×C₄	—	8	2	no	—
363	C₂×C₄	—	8	2	no	—
364	C₂×C₄	—	8	2	no	—
365	C₂×C₄	—	8	2	no	—
366	C₂×C₄	—	8	2	no	—
367	C₂×C₄	—	8	2	no	—
368	C₂×C₄	—	8	2	no	—
369	C₂×C₄	—	8	2	no	—
370	C₂×C₄	#107×C₁₀₉	1	4	yes	complemented
371	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
372	C₂×C₄	#117×C₁₀₉	1	1	yes	characteristic
373	C₂×C₄	C₄×D₄×C₁₀₉	1	2	yes	—
374	C₂×C₄	C₄×Q₈×C₁₀₉	1	2	yes	—
375	C₂×C₄	#120×C₁₀₉	1	1	yes	characteristic
376	C₂×C₄	#121×C₁₀₉	1	2	yes	—
377	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
378	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
379	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	1	yes	characteristic
380	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	1	yes	characteristic
381	C₂×C₄	#123×C₁₀₉	1	1	yes	characteristic
382	C₂×C₄	#123×C₁₀₉	1	2	yes	—
383	C₂×C₄	#124×C₁₀₉	1	1	yes	characteristic
384	C₂×C₄	#125×C₁₀₉	1	2	yes	—
385	C₂×C₄	#125×C₁₀₉	1	2	yes	—
386	C₂×C₄	#126×C₁₀₉	1	2	yes	—
387	C₂×C₄	C₄⋊Q₈×C₁₀₉	1	1	yes	characteristic
388	C₂×C₄	C₄⋊Q₈×C₁₀₉	1	2	yes	—
389	C₈	—	2	4	no	—
390	C₈	—	4	2	no	—
391	C₂²	#380×C₁₀₉	1	1	yes	characteristic
392	C₂²	#389×C₁₀₉	1	2	yes	—
393	C₂²	#394×C₁₀₉	1	1	yes	characteristic
394	C₂²	#396×C₁₀₉	1	2	yes	—
395	C₂²	#398×C₁₀₉	1	1	yes	characteristic
396	C₂²	C₂×#107×C₁₀₉	1	1	yes	characteristic
397	C₂²	#427×C₁₀₉	1	1	yes	characteristic
398	C₂²	#428×C₁₀₉	1	2	yes	—
399	C₂²	#459×C₁₀₉	1	2	yes	—
400	C₂²	#462×C₁₀₉	1	2	yes	—
401	C₂²	#463×C₁₀₉	1	2	yes	—
402	C₂²	#464×C₁₀₉	1	2	yes	—
403	C₂²	#466×C₁₀₉	1	1	yes	characteristic
404	C₂²	#466×C₁₀₉	1	1	yes	characteristic
405	C₂²	#467×C₁₀₉	1	1	yes	characteristic
406	C₂²	#467×C₁₀₉	1	1	yes	characteristic
407	C₂²	#468×C₁₀₉	1	2	yes	—
408	C₂²	#469×C₁₀₉	1	1	yes	characteristic
409	C₂²	#470×C₁₀₉	1	1	yes	characteristic
410	C₂²	#474×C₁₀₉	1	2	yes	—
411	C₂²	#477×C₁₀₉	1	2	yes	—
412	C₂²	#478×C₁₀₉	1	2	yes	—
413	C₂²	#479×C₁₀₉	1	2	yes	—
414	C₄	—	2	4	no	—
415	C₄	—	4	2	no	—
416	C₄	—	8	4	no	—
417	C₄	—	8	4	no	—
418	C₄	#386×C₁₀₉	1	1	yes	characteristic
419	C₄	#387×C₁₀₉	1	2	yes	—
420	C₄	#391×C₁₀₉	1	1	yes	characteristic
421	C₄	#395×C₁₀₉	1	1	yes	characteristic
422	C₄	#398×C₁₀₉	1	2	yes	—
423	C₄	#400×C₁₀₉	1	1	yes	characteristic
424	C₂	C₁₀₉×#1441	1	1	yes	characteristic
425	C₂	C₁₀₉×#1808	1	1	yes	characteristic
426	C₂	C₁₀₉×#1809	1	1	yes	characteristic
427	C₂	C₁₀₉×#1947	1	2	yes	—
428	C₂	C₁₀₉×#1949	1	2	yes	—
429	C₂	C₁₀₉×#2054	1	1	yes	characteristic
430	C₂	C₁₀₉×#2056	1	1	yes	characteristic
431	C₂	C₁₀₉×#2067	1	2	yes	—
432	C₂	C₁₀₉×#2068	1	2	yes	—
433	C₂	C₁₀₉×#2086	1	1	yes	characteristic
434	C₂	C₁₀₉×#2087	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₀₉	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂³×C₄	C₂³×C₁₀₉
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₀₉ ◁ C₂²×C₄²×C₁₀₉ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
15. C₂×C₄⋊C₈×C₁₀₉	C₄	semidirect
152. C₂×C₄×C₁₀₉	#107	semidirect
206. #11232	C₁₀₉	direct (N × H)
223. C₁₀₉	#11232	direct (N × H)
233. C₂×C₄⋊C₈	C₄×C₁₀₉	semidirect
370. C₂×C₄	#107×C₁₀₉	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#107×C₁₀₉	2
C₂²×C₄⋊C₈×C₁₀₉	2
C₂×#340×C₁₀₉	2
C₁₀₉×C₄⋊#95	2
#11232	109

Subgroups — C109×#11232 #42129226

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#11232 #42129226